Momentum ji ji Filtered i i i i - PowerPoint PPT Presentation

Outline  Filtering  Large Scales and Subgrid Scales  Subgrid Scales Stresses  Leonard Stress  Smogorinski Model  Cross Stress ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi   Decomposition i  i  Large Scale Subgrid Scale v v Mass 0 0                     x x t t t t t t p p p p p p v v v v v v i i ij ij ij ij ij ij i i i i i i Filter (Gaussian)           t t v v v v                             Momentum ji ji Filtered       i i i i       v v f f     x x G G x x , , x x x x d d x x       j j i i t t x x x x     Large Scale j j j j D D                 Newtonian t t p p v v v v     Note That       0 0 ij ij ij ij i i , , j j j j , , i i ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1

                S S R R L L C C     i  i  S S t t t t t t t t v v t t t t v v v v                 ji ji ji ji ji ji ji ji ji ji ji ji 0 0 i i i i v v f f               j j i i x x t t x x x x x x     i i j j j j j j Reynolds           R R t t v v v v Stresses ji ji j j i i     Subgrid-Scale Stress Tensor Leonard         L L     t t v v v v v v v v Stresses ji ji j j i i j j i i             S S t t v v v v v v v v Cross ji ji j j i i j j i i             C C t t v v v v v v v v Stresses ji ji j j i i i i j j ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi                         R R v v c c t t 2 2 D D v v 1 1     t t c c v v v v v v v v     j j i i ji ji T T ji ji D D     ji ji r r j j i i j j i i     ji ji 2 2 x x x x     i i j j c r  c r  1 Speziale 1 Smagorinsky      2  2 Galilean Invariance 1 1         2 2 c c D D D D Model     T T S S k k k k Leonard Stresses are Evaluated Grid Size Smagorinsky Constant     c S  c S     3  3 1 1           0 0 . . 21 21         L L t t v v v v v v v v 1 1 2 2 3 3 ji ji j j i i j j i i ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2

ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi (a) (b) ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3

 LES captures computes for the large eddies of turbulence while the sub-grid scales are modeled.  LES can provide more accurate description of turbulent flows.  Smagorinski model provide a reasonable approximation. ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 4